Can a frame of reference be wrong in determining motion in space-time?

[selfAdjoint]

The question is: can we have a coordinate system of space-time where all geodesics can be represented as straight lines.
To determine if a geodesic is straight that passes a warped region in space-time we would need a description of the curvature.
That's if why we cannot conclude that if dx/dt, dy/dt and dz/dt remains constant we have a straight line. Or in other words without a metric describing the warping at all places on the wordline we would not be able to conclude that.

This is exactly right. I have bolded the key statement. And in (pseudo-)Riemannian geometry we have a metric tensor from which, in this case, we can derive a connection, a mathematical expression in the partial derivatives of the metric tensor, from which we get a covariant derivative which finally gives us a Riemannian or Curvature tensor, which describes the curvature. All of these derivations are straight computations from the metric; that is a feature of Riemannian geometry. Once we have this mathematical machinery that determines the curvature, we can do further mathematical derivations to find the form of the geodesic equations.

Now if the Riemann tensor is not identically zero then the form of the geodesic equations is not linear. Translation into English; if spacetime has nonzero curvature somewhere, then the geodesics there cannot be straight lines. This again is a straight computation and so it really only depends on the metric. In a Riemannian geometry the metric determines the curvature, and the geodesics for non zero curvature are not linear

 

[MeJennifer]

if spacetime has nonzero curvature somewhere, then the geodesics there cannot be straight lines.

Why not?
What else is a geodesic but a straight line on a curved surface?

 

[MeJennifer]

if spacetime has nonzero curvature somewhere, then the geodesics there cannot be straight lines.

Then what do you consider a straight line on a curved surface?
What else is a geodesic but a straight line on a curved surface?

 

[selfAdjoint]

Why not?
What else is a geodesic but a straight line on a curved surface?

A path which has minimum curvature between its endpoints.

 

[MeJennifer]

A path which has minimum curvature between its endpoints.

And you say that calling that a straight line is wrong?

 

[selfAdjoint]

And you say that calling that a straight line is wrong?

A straight liine is one where the local tangent vector to it at some point stays parallel to itself as you move along the line. In a curved geometry that can't happen; there are no mathematically straight lines in a curved Riemannian geometry that can serve for geodesics.

 

[MeJennifer]

A straight liine is one where the local tangent vector to it at some point stays parallel to itself as you move along the line. In a curved geometry that can't happen; there are no mathematically straight lines in a curved Riemannian geometry that can serve for geodesics.

You basically limit the concept of a straight line to Euclidean surfaces only.

But anyway I learned my lesson, no more straight line when I can use geodesic.

 

[nrqed]

You basically limit the concept of a straight line to Euclidean surfaces only.

But anyway I learned my lesson, no more straight line when I can use geodesic.

The whole discussion arose because you had decided to make you own definition of "straight lines" as being the same as geodesics, without making first the effort to learning what the rest of the physics community defines as a straight line. Sure, you can do that if you want, but then the price to pay is endless discussions like this. And this is only because you did not pause and ask, humbly, "I always thought that a straight line is the same as a geodesic. Can someone confirm this or correct me?"



SelfAdjoint has given the standard definitions of straight lines and geodesics. Two-dimensional beings living on on the surface of a sphere could realize they live on a cruved surface using local measurements only and that would be based on the fact that geodesics would *not* be straight lines. For example by using parallel transport of a vector along a short geodesic, turning by 90 degrees and so on until one is back to the starting point after having gone through a paralleliped. The vector would come out rotated which would be an indication that the geodesics were not straight lines (according to the definition used by the physics community). The amount of rotation would allow the calculation of the curvature.

What did *you* call a line for which the tangent vector remains parallel to itself?

You have to be willing to learn the terminology used by everyone before arguing that they are wrong about something.

Regards

 

[nrqed]

You basically limit the concept of a straight line to Euclidean surfaces only.

But anyway I learned my lesson, no more straight line when I can use geodesic.

He did not "limit" the concept of straight line to Euclidiean spaces only, he gave it a rigorous mathematical definition, which is the one used by everyone in the field. The fact that in a curved Riemannian geometry straight lines do not correspond to geodesics follow from their mathematical defintion.

One could as well argue that *you* had restricted (in a completely different way) the meaning of straight lines too! You had restricted them to be identical to geodesics! The main difference with SelfAdjoint's "limitation" is that his corresponds to what everyone uses in the field.

Regards

Patrick

 

[JesseM]

The question is: can we have a coordinate system of space-time where all geodesics can be represented as straight lines.
To determine if a geodesic is straight that passes a warped region in space-time we would need a description of the curvature.
That's if why we cannot conclude that if dx/dt, dy/dt and dz/dt remains constant we have a straight line.

Sigh. I made it very clear throughout this entire thread that when I said you couldn't find a coordinate system where all geodesics were "straight in the coordinate sense", I was defining the term "straight in the coordinate sense" to mean constant dx/dt, dy/dt, and dz/dt. I repeated this over and over again in many posts, just to make sure there was no confusion on this point. So if you are conceding that you cannot find a coordinate system where all geodesics have constant dx/dt, dy/dt and dz/dt, regardless of whether this would disqualify them from being "straight lines" under your preferred definition, then you are either admitting you were wrong all along in disagreeing with me, or admitting that you were not even paying a bare minimum of attention to what I was actually saying in my posts (since I repeated this definition in like every other post!)